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README
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README
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This package is almost complete, with the exception of a user guide, which is the intended purpose of this README.
Background and Overview:
The below is a quick overview of this package, though for a thorough background of the mathematics involved, see http://www.sciencedirect.com/science/article/pii/S0024379506000218
This collection of Python files was written so that I, or others, could use some of the algorithms described in "Bijective Matrix Algebra" written by Loehr and Mendes (2006). In their paper, they explored a kind of abstraction of doing algebra over a multivariate ring of polynomials over the integers. Specifically, they defined the notion of combinatorial scalars as sets of objects, each of which has a sign and weight (in terms of a monomial with coefficient of 1). And rather than defining equality of scalars to be set equality, it discusses certain mappings whereby scalars can "reduce" to other scalars (i.e. if their generating functions are the same).
At any rate, their paper concerned the notion of matrix identities where the entries in the matrices are these combinatorial scalars. The specific identities discussed corresponds to a collection of involutions and bijections showing that the product of matrices A and B, AB reduce to I (identity matrix).
There are many situations where a combinatorial argument can be given to show that AB reduces to I, but it is unknown whether BA reduces to I.
In their paper, they describe an deterministic algorithm whereby they construct the maps necessary to show that BA reduces to I, given the set of maps which show that BA reduces to I.
This package contains most of the useful constructs in their paper and can therefore be used to provide insight into certain open problems where a combinatorial argument is given for AB reducing to I, but not for BA reducing to I.
Incidentally, Loehr and Mendes' solved a problem posed by Eğecioğlu and Remmel in 1990 about the Kotska matrix and its inverse in 1990. There was a combinatorial proof for one ordering but the other was not published until 2006.
Problem Statement:
Is this process an involution?
That is, given that AB reduces to I (where a collection of maps are given as a part of using the word "reduces"), if one applies the process outlined in the paper, they get that BA reduces to I (along with newly canonically constructed maps). Now, if the process is applied once more, will we always get the same collection of maps with which we started?
Preliminary Results?
There seems to be certain cases where this process falls into a loop after a couple iterations, but it seems that when taking on a specific example of Stirling Matrices using an arbitrary collection of maps, we see that the Loehr-Mendes bijection is not an involution.
More work will be done prior to writing a paper, which will include understanding what how minor changes in the maps showing that AB reduces to I affect the new collection of maps for BA reduces to I.
Technical Stuff: to get this package working on your sage installation:
Add "'sage.bijectivematrixalgebra'," to: devel/sage/setup.py
Add folder "bijectivematrixalgebra" to: devel/sage/sage
Add contents of this repository to the "bijectivematrixalgebra" folder.
Run the sage -b command to build this repository.
User Guide:
See SampleOutput.txt for a use of some of the basic commands.
More to come here soon...